Graph-Based Transform for 2D Piecewise Smooth Signals with Random Discontinuities

The graph-based transform has recently emerged as an effective tool for compressing some special signals such as depth images in 3D videos. However, one limitation of this approach is that it needs to apply eigen-decomposition to each block. To reduce the complexity, in this paper, we develop a systematic approach to find a universal optimal graph-based transform for a class of 2D piecewise smooth signals. Each block in the class can include a discontinuity whose locations in different rows are randomly distributed within a confined region. We first define a special 2D graph model for this class of signals. Our derivation then reveals that the inverse of the covariance matrix of this class of signals is equal to its graph Laplacian with a bias value added to the first diagonal element. Furthermore, the edge values within the confined region have a closed-form expression. If the bias value is assumed negligible then the approximation of the optimal transform for the class of signals is given by the eigenvectors of the true graph Laplacian and can be pre-computed. Therefore online eigen-decomposition for the class of signals can be avoided, and the complexity of the encoder and decoder can thus be reduced. The feasibility of the proposed scheme is demonstrated via depth image coding examples.